Why are dogs "neutered", horses "gelded", and people "castrated"?

Castrate is a general term that can be applied both to human beings and animals. Geld and neuter are respectively used for horses and pets probably because they were originally mainly used to refer to those animals: See etymology below:

Castrate: (v.)

• 1610s (implied in castrated), back-formation from castration (q.v.), or from Latin castratus, past participle of castrare.

Geld (v.):

• "to castrate," c. 1300, from Old Norse gelda "to castrate," said in Watkins to be from Proto-Germanic *galdjan "to castrate," from PIE *ghel- (3) "to cut." Related to other words which, if the derivation is correct, indicate a general sense of "barren." Compare Old Norse geld-fe "barren sheep" and geldr (adj.) "barren, yielding no milk, dry,"which yielded Middle English geld "barren" (of women and female animals); also Old High German galt "barren," said of a cow.
• Gelding (n.): late 14c., "castrated animal" (especially a horse), also "a eunuch" (late 13c. as a surname), from Old Norse geldingr "wether; eunuch," from gelda "castrate"

Neuter (v.):

• 1903, from neuter (adj.). Originally in reference to pet cats,

Neutering a male animal, you remove the testicles and leave the sac. To castrate means the whole thing is removed like on a gelding (castrated male horse). Neutering can be applied to either sex - it means removing the sex organs. Castrating means removing the testicles, so it refers to males only. Removing the uterus is called spaying.

Generally, the word neuter is being used for dogs for so long by people, but it doesn't mean that you can't use it for a horse or man. Here, the difference is of the process, whether it's on a man, dog or horse. The word can be changed from place to place. But neuter is generally not used for a human and to use castrate for females isn't a great idea.